Definition:Sigma-Algebra Generated by Collection of Subsets

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Definition

Let $X$ be a set.

Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.


Definition 1

The $\sigma$-algebra generated by $\GG$, $\map \sigma \GG$, is the smallest $\sigma$-algebra on $X$ that contains $\GG$.

That is, $\map \sigma \GG$ is subject to:

$(1): \quad \GG \subseteq \map \sigma \GG$
$(2): \quad \GG \subseteq \Sigma \implies \map \sigma \GG \subseteq \Sigma$ for any $\sigma$-algebra $\Sigma$ on $X$


Definition 2

The $\sigma$-algebra generated by $\GG$, $\map \sigma \GG$, is the intersection of all $\sigma$-algebras on $X$ that contain $\GG$.


Generator

One says that $\GG$ is a generator for $\map \sigma {\GG}$.

Also, elements $G$ of $\GG$ may be called generators.


Also denoted as

Variations of the letter "$M$" can be seen for the $\sigma$-algebra generated by $\GG$:

$\map \MM \GG$
$\map {\mathscr M} \GG$


Also see


Sources